Contents vii §6.5. Recommended reading 167 §6.6. Exercises 168 Chapter 7. Review and Preview 171 §7.1. Introduction 171 §7.2. Necessary conditions 172 §7.3. Sufficient conditions 173 §7.4. Two dependent variables 173 §7.5. History and preview 175 §7.6. Recommended reading 176 Chapter 8. The Homogeneous Problem 177 §8.1. Integrals in parametric form 177 §8.2. Euler–Lagrange equations 180 §8.3. The Weierstrass equation 182 §8.4. Case study: The parametric Queen Dido problem 186 §8.5. Recommended reading 189 §8.6. Exercises 190 Chapter 9. Variable-Endpoint Conditions 191 §9.1. Natural boundary conditions 191 §9.2. Transversality conditions 197 §9.3. Focal points 206 §9.4. Case study: Neile’s parabola 206 §9.5. Recommended reading 211 §9.6. Exercises 212 Chapter 10. Broken Extremals 215 §10.1. The Weierstrass–Erdmann corner conditions 215 §10.2. Carath´ eodory’s indicatrix 224 §10.3. Recommended reading 231 §10.4. Exercises 232
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